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Certain species of plants can benefit from synergistic effects with plant growth-promoting rhizobacteria (PGPR) that improve plant growth and metal accumulation, mitigating toxic effects on plants and increasing their tolerance to heavy metals. The application of PGPR as biofertilizers and atmospheric nitrogen fixators contributes considerably to the intensification of the phytoremediation process. In this paper, we have built a system consisting of rhizospheric Azotobacter microbial populations and Lepidium sativum plants, growing in solutions containing heavy metals in various concentrations. We examined the ability of the organisms to grow in symbiosis so as to stimulate the plant growth and enhance its tolerance to Cr(VI) and Cd(II), to ultimately provide a reliable phytoremediation system. The study was developed at the laboratory level and, at this stage, does not assess the inherent interactions under real conditions occurring in contaminated fields with autochthonous microflora and under different pedoclimatic conditions and environmental stresses. Azotobacter sp. bacteria could indeed stimulate the average germination efficiency of Lepidium sativum by almost 7%, average root length by 22%, average stem length by 34% and dry biomass by 53%. The growth of L. sativum has been affected to a greater extent in Cd(II) solutions due its higher toxicity compared to that of Cr(VI). The reduced tolerance index (TI, %) indicated that plant growth in symbiosis with PGPR was however affected by heavy metal toxicity, while the tolerance of the plant to heavy metals was enhanced in the bacteria-plant system. A methodology based on artificial neural networks (ANNs) and differential evolution (DE), specifically a neuro-evolutionary approach, was applied to model germination rates, dry biomass and root/stem length and proving the robustness of the experimental data. The errors associated with all four variables are small and the correlation coefficients higher than 0.98, which indicate that the selected models can efficiently predict the experimental data.

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Artificial neural networks (ANNs) are comparatively straightforward to understand and use in the analysis of scientific data. However, this relative transparency may encourage their use in an uncritical, and therefore possibly unproductive, fashion. The geometry of a network is among the most crucial factors in the successful deployment of network tools; in this review, we cover methods that can be used to determine optimum or near-optimum geometries. These methods of determining neural network architecture include the following: (i) trial and error, in which architectures chosen semirandomly are tested and modified by the user; (ii) empirical or statistical methods, in which an ANN's internal parameters are adjusted based on the model's performance; (iii) hybrid methods, such as fuzzy inference; (iv) constructive and/or pruning algorithms, that add and/or remove neurons or weights from an initial architecture, respectively, based on a predefined link between architecture and ANN performance; (v) evolutionary strategies, which search the topology space using genetic operators to vary the neural network parameters. Several case studies illustrate the development of neural network models for applications in chemistry and chemical engineering.

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